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User blog:B1mb0w/The Alpha Function
'The Alpha Function' The Alpha Function uses Version 8 code and my S Function (substitution function) to generate every finite integer up to a very large number. The Alpha Function has a growth rate faster than \(f_{LVO}(n)\) for any n. The Alpha Function has one parameter: \(\alpha®\) where r is any real number. The real number is manipulated by Sequence Generator Code (see references) to create a finite sequence of finite integers. The sequences can be translated to represent every unique finite integer (up to the size of \(f_{LVO}(n)\) for any n). See the references section for Granularity Examples of the Alpha Version 8 blog that illustrate how finely tuned the functions are to access any and every finite integer (up to the size of \(f_{LVO}(n)\) for any n). 'What is the Alpha Function' My motivation to create this function was to develop a finely grained number notation system for really big numbers. \(\alpha(2)\) for example can be used to reference the number 8. Therefore 2 is the Alpha Index for the number 8. Alpha needs to reference big numbers very quickly to be useful, therefore it uses The S Function for this purpose. Alpha should also be monotonically increasing and every input real \(a > b\), results in a larger output number, where \(\alpha(a) >= \alpha(b)\) in all cases. The function is finely grained. It accepts a real number input and depending upon the precision of the real number, it can locate any and every big number up to the size of \(f_{LVO}(v)\) for any n. The Alpha Function therefore has a growth rate greater than the Large Veblen Ordinal (LVO). 'High Level Description' The Alpha Function is 'calibrated' to accept real number inputs up to 10,000 and generate unique S() function outputs representing any and every big number up to the size of \(f_{LVO}(v)\) for any n. Refer to my Alpha Version 8 blog for more information. 'Some Calculations' Refer to my Version 8 blog for more examples: \(\alpha(0.00) = S(0,0,0) = 0\) \(\alpha(1.00) = S(1,0,0) = 1\) \(\alpha(2.00) = S(2,T(0),1) = f_2(2) = 8\) \(\alpha(e) = \alpha(2.71828182845905) = S(S(2,T(0),1),0,S(2,0,1)) = 11\) \(\alpha(3.00) = S(S(2,T(0),1),1,1) = f_1(f_{\omega}(2)) = 16\) \(\alpha(\pi) = \alpha(3.14159265358979) = S(S(S(2,T(0),1),1,1),1) + 1 = 17\) \(\alpha(4.00) = S(2,T(0) + 1,1) = f_{\omega + 1}(2) = 2048\) \(\alpha(4.891) = S(S(2,T(0) + 1,1),1,9) = 2048.2^9 = 1048576 >>\) One Million \(\alpha(4.992035) = S(S(2,T(0) + 1,1),1,S(8,1,5) + S(8,1,3) + 2)\) \(= 2048.2^{8.2^5 + 8.2^3 + 2} = 2048.2^{246 + 64 + 2} = 17.10^{99} >>\) Googol \(\alpha(\) TBA \() = f_2^4(4) = f_3(4) >>\) Googolplex \(\alpha(\) TBA \() = f_3^3(4) >> g_1\) where \(g_{64} = G\) is Graham's number \(\alpha(\) TBA \() = f_{\omega + 1}(64) >> g_{64} = G\) is Graham's number 'tree and TREE Functions' The Alpha function grows faster than \(f_{LVO}(n)\) for any n. This means it is faster than the TREE function. Refer to my blogs on The S Function and Comparison Table of Ordinal Collapsing Functions for more information. 'Ruler Functions' I have started work on a set of Ruler Functions based on the Alpha function that will be useful to measure the size of very large numbers. A rough sketch of how this will look is: \(\alpha_0(100) = \alpha(\) TBA \() = f_{SVO}(3)\) approximately \(\alpha_1(100) = \alpha(\) TBA \() = f_{\omega + 1}(64) >> g_{64} = G\) is Graham's number \(\alpha_2(100) = \alpha(\) TBA \() = f_3(4) >>\) Googolplex \(\alpha_3(100) = \alpha(\) TBA \() = f_2^2(6) >>\) Googol These functions will be calibrated to align with some famous googology numbers, so that 100 on each ruler will equal one of these famous numbers (approximately). The \(\alpha_3\) function is intended to be identical to Scientific Notation (in base 10), therefore the following will be true by definition: \(\alpha_3(100) = 10^{100} =\) Googol WORK IN PROGRESS 'Sequence Generator Code' The Alpha Function uses Sequence Generator Code to create a finite sequence of finite integers that can be translated into a unique combination of Verben ordinals and FGH functions. See references section for more information. 'Comments and Questions' Look forward to comments and questions. Cheers B1mb0w. 'References' The Alpha Function *''The S Function (substitution function)'' **''Version 2'' **''Version 1'' *''Sequence Generator Code (Overview and Syntax)'' **''Version 8'' **''Version 7'' *''Finite: a tour of the finite numbers'' *''Comparison Table of Ordinal Collapsing Functions'' The following Out-Dated References are no longer relevant because the Alpha Function has been changed completely and is based on different function and program logic. Please keep this in mind if you refer to any of these blogs. *''Fundamental Sequences'' **''FGH Function with Omega'' *''The Beta Function'' *''Sequence Generator Code (Overview and Syntax)'' **''Version 6'' *''The J Function'' *''Calculating Alpha Numbers'' *''Sequence Generator Code (Overview and Syntax)'' **''Version 5 (using Sequence Generator code)'' **''Version 4 (using Sequence Generator code)'' *''Program Code'' **''Version 3'' **''Version 2'' **''Version 1'' *''The previous Alpha Function'' *''The (old) Alpha Function'' Category:Blog posts